Complex division, real part

Average Error: 41.47% → 15.65%

Time: 12.3s

Precision: binary64

Cost: 20560

Math

\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-190}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))))
   (if (<= c -1.5e+80)
     (+ (/ a c) (/ (* (/ b c) d) c))
     (if (<= c -5.8e-190)
       t_0
       (if (<= c 5.8e-125)
         (* (/ 1.0 d) (+ b (/ a (/ d c))))
         (if (<= c 3.3e+72) t_0 (+ (/ a c) (/ (/ d c) (/ c b)))))))))

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

↓

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	double tmp;
	if (c <= -1.5e+80) {
		tmp = (a / c) + (((b / c) * d) / c);
	} else if (c <= -5.8e-190) {
		tmp = t_0;
	} else if (c <= 5.8e-125) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (c <= 3.3e+72) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((d / c) / (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

↓

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)))
	tmp = 0.0
	if (c <= -1.5e+80)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b / c) * d) / c));
	elseif (c <= -5.8e-190)
		tmp = t_0;
	elseif (c <= 5.8e-125)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (c <= 3.3e+72)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) / Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.5e+80], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b / c), $MachinePrecision] * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.8e-190], t$95$0, If[LessEqual[c, 5.8e-125], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.3e+72], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	41.47%
Target	0.75%
Herbie	15.65%

\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if c < -1.49999999999999993e80

   1. Initial program 60.16

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 28.07

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]

   3. Simplified 18.81

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}} \]
      Proof

      [Start] 28.07	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      *-commutative [<=] 28.07	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      unpow2 [=>] 28.07	\[ \frac{a}{c} + \frac{b \cdot d}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 18.81	\[ \frac{a}{c} + \color{blue}{\frac{b}{c} \cdot \frac{d}{c}} \]

   4. Applied egg-rr 18

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{b}{c} \cdot d}{c}} \]

   if -1.49999999999999993e80 < c < -5.8000000000000004e-190 or 5.8000000000000004e-125 < c < 3.3e72

   1. Initial program 24.89

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Applied egg-rr 16.19

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}} \]

   if -5.8000000000000004e-190 < c < 5.8000000000000004e-125

   1. Initial program 37.47

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around 0 16.45

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}} \]

   3. Simplified 13.02

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}} \]
      Proof

      [Start] 16.45	\[ \frac{b}{d} + \frac{c \cdot a}{{d}^{2}} \]
      *-commutative [<=] 16.45	\[ \frac{b}{d} + \frac{\color{blue}{a \cdot c}}{{d}^{2}} \]
      unpow2 [=>] 16.45	\[ \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]
      times-frac [=>] 13.02	\[ \frac{b}{d} + \color{blue}{\frac{a}{d} \cdot \frac{c}{d}} \]

   4. Applied egg-rr 11.97

      \[\leadsto \color{blue}{\frac{1}{d} \cdot \left(\frac{a}{d} \cdot c + b\right)} \]

   5. Applied egg-rr 10.73

      \[\leadsto \frac{1}{d} \cdot \left(\color{blue}{\frac{a}{\frac{d}{c}}} + b\right) \]

   if 3.3e72 < c

   1. Initial program 58.32

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf 25.7

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}} \]

   3. Simplified 17.83

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}} \]
      Proof

      [Start] 25.7	\[ \frac{a}{c} + \frac{d \cdot b}{{c}^{2}} \]
      *-commutative [<=] 25.7	\[ \frac{a}{c} + \frac{\color{blue}{b \cdot d}}{{c}^{2}} \]
      unpow2 [=>] 25.7	\[ \frac{a}{c} + \frac{b \cdot d}{\color{blue}{c \cdot c}} \]
      times-frac [=>] 17.83	\[ \frac{a}{c} + \color{blue}{\frac{b}{c} \cdot \frac{d}{c}} \]

   4. Applied egg-rr 17.72

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{\frac{d}{c}}{\frac{c}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 15.65

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.02%
Cost	1752

\[\begin{array}{l} t_0 := \frac{b \cdot d + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -3.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.12 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 640000000000:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

Alternative 2
Error	25.08%
Cost	1233

\[\begin{array}{l} t_0 := \frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-75} \lor \neg \left(c \leq 980000000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]

Alternative 3
Error	24.91%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;c \leq -1.06 \cdot 10^{+65}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-75} \lor \neg \left(c \leq 900000000000\right):\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]

Alternative 4
Error	24.86%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+64}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-75}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq 580000000000:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

Alternative 5
Error	24.9%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;c \leq -1.06 \cdot 10^{+65}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 1.36 \cdot 10^{-76}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq 700000000000:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

Alternative 6
Error	25.28%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+65}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1450000000000:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{c}}{\frac{c}{b}}\\ \end{array} \]

Alternative 7
Error	29.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+65} \lor \neg \left(c \leq 1.4 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]

Alternative 8
Error	29.52%
Cost	968

\[\begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 36000000000000:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9
Error	36.12%
Cost	456

\[\begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10
Error	58.21%
Cost	192

\[\frac{a}{c} \]

Reproduce

herbie shell --seed 2023121 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))